Vol. 16, No. 1 (2010), pp. 1–7.
EQUIVALENCE OF
n-NORMS ON THE SPACE OF
p-SUMMABLE SEQUENCES
Anwar Mutaqin
1and Hendra Gunawan
21Department of Mathematics Education, Universitas Sultan Ageng
Tirtayasa, Serang, Indonesia, anwarmutaqin@gmail.com
2Department of Mathematics, Institut Teknologi Bandung, Bandung,
Indonesia, hgunawan@math.itb.ac.id
Abstract. We study the relation between two knownn-norms onℓp, the space of p-summable sequences. Onen-norm is derived from G¨ahler’s formula [3], while the other is due to Gunawan [6]. We show in particular that the convergence in one
n-norm implies that in the other. The key is to show that the convergence in each of thesen-norms is equivalent to that in the usual norm onℓp.
Key words:n-normed spaces,p-summable sequence spaces,n-norm equivalence.
Abstrak. Dalam makalah ini dipelajari kaitan antara dua norm-ndiℓp, ruang barisansummable-p. Norm-npertama diperoleh dari rumus G¨ahler [3], sementara norm-nkedua diperkenalkan oleh Gunawan [6]. Ditunjukkan antara lain bahwa kekonvergenan dalam norm-nyang satu mengakibatkan kekonvergenan dalam
norm-nlainnya. Kuncinya adalah bahwa kekonvergenan dalam masing-masing norm-n
tersebut setara dengan kekonvergenan dalam norm biasa diℓp.
Kata kunci: ruang norm-n, ruang barisansummable-p, kesetaraan norm-n
1. Introduction
In [6], Gunawan introduced an n-norm on ℓp (1 ≤ p ≤ ∞), the space of p-summable sequences (of real numbers), given by the formula
kx1, . . . , xnkp:=
1 n!
X
j1
· · ·X
jn abs
¯ ¯ ¯ ¯ ¯ ¯ ¯
x1j1 · · · xnj1
..
. . .. ... x1jn · · · xnjn
¯ ¯ ¯ ¯ ¯ ¯ ¯
p
1/p
2000 Mathematics Subject Classification: 46B15 Received: 31-03-2010, accepted: 05-08-2010.
for 1≤p <∞, and
Rwhich satisfies the following four conditions:
(N1)kx1, . . . , xnk= 0 if and only ifx1, . . . , xn are linearly dependent;
(N2)kx1, . . . , xnk is invariant under permutation;
(N3)kαx1, . . . , xnk=|α| kx1, . . . , xnk forα∈R;
(N4)kx1+x′1, x2, . . . , xnk ≤ kx1, x2, . . . , xnk+kx′1, x2, . . . , xnk.
The theory ofn-normed spaces was developed by G¨ahler in 1969 and 1970 [3, 4, 5]. The special case where n = 2 was studied earlier, also by G¨ahler, in 1964 [2]. Related work may be found in [1]. For more recent works, see [7, 8, 10].
IfX is equipped with a norm k · k, then according to G¨ahler, one may define ann-norm onX (assuming thatX is at leastn-dimensional) by the formula
kx1, . . . , xnk∗:= sup
HereX′denotes the dual ofX, which consists of bounded linear functionals onX. ForX =ℓp (1 ≤p <∞), we know that X′ =ℓp′
with 1p+ p1′ = 1. In this
case the above formula reduces to
kx1, . . . , xnk∗p:= sup
Thus, onℓp, we have two definitions of n-norms, one is due to Gunawan and the
other is derived from G¨ahler’s formula. For p = 2, one may verify that the two n-norms are identical.
The purpose of this paper is to study the relation between the twon-norms onℓpfor 1≤p <∞. In particular, we shall show that the twon-norms are weakly
a sequence (x(m)) in ann-normed space (X,k·, . . . ,·k) is said toconvergetox∈X ifkx(m)−x, x2, . . . , xnk →0 asm→ ∞, for everyx2, . . . , xn ∈X.
For convenience, we prove the result forn = 2 first, and then extend it to anyn≥2.
2. Main Results
Recall that Gunawan’s definition of 2-norm onℓp (1≤p≤ ∞) is given by
Meanwhile, G¨ahler’s definition is given by
kx, yk∗
By the same trick as in [6], one may obtain
kx, yk∗p= sup
From the last expression, we have the following fact.
Fact 2.1. The inequalitykx, yk∗
p≤21/pkx, ykp holds for everyx, y∈ℓp.
Proof. By H¨older’s inequality for 1
Now, observe that
This proves the inequality.
Note that forp= 1, H¨older’s inequality gives
1
Corollary 2.2 If (x(m))converges in k·,·kp, then it also converges (to the same
limit) ink·,·k∗ p.
We shall show next that the convergence ink·,·k∗
palso implies the convergence
in k·,·kp. We do so by showing that: (1) the convergence ink·,·k∗p implies that in
k · kp, and (2) the convergence ink · kp implies that ink·,·kp.
The second implication is already proved in [6] (using the inequalitykx, ykp≤
21−(1/p)kxk
pkykp). Hence it remains only to show the first implication.
Theorem 2.3 If (x(m)) converges in k·,·k∗
p, then it also converges (to the same
withzj :=sgn(xj(m)−xj)|xj(m)−xj|
p−1
kx(m)−xkp−1
p and
w:= (1,0,0, . . .), then we have
∞
X
j=2
|xj(m)−xj|p
kx(m)−xkpp−1
< ǫ.
[Here we are handling only the case where kx(m)−xkp 6= 0.] Next, if we take
y := (0,1,0, . . .), z= (z1,0,0, . . .) withz1:= sgn(x1(m)−x1)|x1(m)−x1| p−1
kx(m)−xkp−1
p and
w:= (0,1,0, . . .), then we have
|x1(m)−x1|p
kx(m)−xkpp−1
< ǫ.
Adding up, we get
kx(m)−xkp= ∞
X
j=1
|xj(m)−xj|p
kx(m)−xkpp−1
<2ǫ.
This shows that (x(m)) converges toxin k · kp. ¤
Corollary 2.4A sequence is convergent ink·,·k∗p if and only if it is convergent (to
the same limit) ink·,·kp.
All these results can be extended ton-normed spaces for any n≥2. As an extension of Fact 2.1, we have:
Fact 2.5The inequalitykx1, . . . , xnk∗p≤(n!)1/pkx1, . . . , xnkpholds for everyx1, . . . , xn∈ℓp.
Corollary 2.6 If (x(m)) converges in k·, . . . ,·kp, then it converges (to the same
limit) ink·, . . . ,·k∗p.
Analogous to Theorem 2.3, we have:
Theorem 2.7If(x(m))converges ink·, . . . ,·k∗
p, then it also converges (to the same
limit) ink · kp.
Proof. Let (x1(m)) be a sequence inℓp which converges tox1= (x11, x12, . . .)∈ℓp in k·, . . . ,·k∗p. Then, for anyǫ >0, there exists an N ∈Nsuch that form≥N we
have
1 n!
X
j1
· · ·X
jn
¯ ¯ ¯ ¯ ¯ ¯ ¯
x1j1(m)−x1j1 · · · x1jn(m)−x1jn ..
. . .. ...
xnj1 · · · xnjn
¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯
z1j1 · · · z1jn ..
. . .. ... znj1 · · · znjn
¯ ¯ ¯ ¯ ¯ ¯ ¯
< ǫ
for every x2, . . . , xn ∈ ℓp and z1, . . . , zn ∈ ℓp with kz1k, . . . ,kznk ≤ 1. Now, take
term and z1 = (z11, z12, . . .)∈ ℓp
′
with z1j := sgn(x1j(m)−x1j)|x1j(m)−x1j|
p−1
kx1(m)−x1kp−1
p
, then we have
∞
X
j1=n
|x1j1(m)−x1j1|
p
kx1(m)−x1kpp−1
< ǫ.
Next, if we take xk =zk := (0, . . . ,0,1,0, . . .) for every k = 2, . . . , n, where 1 is
k-th term, and z1:= (z11,0,0, . . .) with z11:= sgn(x11(m)−x11)|x11(m)−x11| p−1
kx1(m)−x1kp−1
p
, then we have
|x11(m)−x11|p
kx1(m)−x1kpp−1
< ǫ.
Similarly, if we alter the position of the entry 1 inxk andzk fork= 2, . . . , n, and
change the nonzero entry ofz1 accordingly, then we can get
|x12(m)−x12|p
kx1(m)−x1kpp−1
< ǫ
and so on until
¯
¯x1(n−1)(m)−x1(n−1)
¯ ¯
p
kx1(m)−x1kpp−1
< ǫ.
Adding up, we get
kx1(m)−x1kp=
∞
X
j1=1
|x1j1(m)−x1j1|p
kx1(m)−x1kpp−1
< nǫ.
This shows that (x(m)) converges toxin k · kp. ¤
Corollary 2.8A sequence is convergent ink·, . . . ,·k∗
p if and only if it is convergent
(to the same limit) ink·, . . . ,·kp.
Related to the above results, one may also prove that a sequence is Cauchy in k·, . . . ,·k∗
p if and only if it is Cauchy in k·, . . . ,·kp. [A sequence (x(m)) in an
n-normed space (X,k·, . . . ,·k) is Cauchy if givenǫ >0 there exists anN ∈Nsuch
that kx(l)−x(m), x2, . . . , xnk< ǫ wheneverl, m≥N, for everyx2, . . . , xn ∈ X.]
Since (ℓp,k·, . . . ,·k
p) is a Banach space [6], we conclude, by Theorem 2.7, that
(ℓp,k·, . . . ,·k∗
p) also forms ann-Banach space.
3. Concluding Remarks
As we have mentioned earlier, the case where p = 2 is of course special. Here, the two n-norms k·, . . . ,·k2 and k·, . . . ,·k∗2 are identical. Indeed, by using Cauchy-Schwarz inequality (see [9]), one may obtain
kx1, . . . , xnk∗2= sup
zi∈ℓ2,kzik2≤1
i= 1,...,n
¯ ¯ ¯ ¯ ¯ ¯ ¯
hx1, z1i · · · hx1, zni
..
. . .. ...
hxn, z1i · · · hxn, zni
¯ ¯ ¯ ¯ ¯ ¯ ¯
By taking z1, . . . , zn to be the orthonormalized vectors obtained from x1, . . . , xn
through Gram-Schmidt process, one can show that the above upper bound is ac-tually attained. Hence we have
kx1, . . . , xnk∗2=kx1, . . . , xnk2.
For p6= 2, things are not so simple and we have difficulties in proving the strong equivalence between the two n-norms k·, . . . ,·k∗
p and k·, . . . ,·kp. The research on
this problem, however, is still ongoing.
Acknowledgement. The research was carried out while the first author did his master thesis at Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung.
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